Transport in Porous Media

, Volume 127, Issue 3, pp 643–660 | Cite as

A Pore-Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments

  • David Landa-MarbánEmail author
  • Na Liu
  • Iuliu S. Pop
  • Kundan Kumar
  • Per Pettersson
  • Gunhild Bødtker
  • Tormod Skauge
  • Florin A. Radu


In this paper, we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided into two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substance (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation, whereas a diffusion–convection equation is involved for the transport of nutrients. At the biofilm–water interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/death of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by ± 10% of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.


Biofilm Numerical simulations Laboratory experiments Microbial enhanced oil recovery Porosity 

List of Symbols


Nutrient concentration


Nutrient diffusion coefficient


Biofilm thickness


Nutrient flux




Bacterial decay rate coefficient


Stress coefficient


Monod half-velocity coefficient


Pore length




Water velocity


Tangential shear stress




Reference water velocity


Velocity of the biomass


Pore width


Growth yield coefficient

Greek Symbols

\(\mu \)

Dynamic viscosity

\(\mu _\mathrm{n}\)

Maximum rate of nutrient utilization

\(\nu \)

Unitary normal vector

\(\nu _n\)

Interface velocity

\(\varPhi \)

Growth velocity potential

\(\rho \)


\(\tau \)

Unitary tangential vector

\(\theta \)

Volume fraction



Active bacteria




Dead bacteria











Arbitrary Lagrangian–Eulerian


Extracellular polymeric substance


Microbial enhanced oil recovery



The work of DLM, NL, KK, PP, GB, TS, and FAR was partially supported by GOE-IP and the Research Council of Norway through the projects IMMENS No. 255426 and CHI No. 255510. ISP was supported by the Research Foundation-Flanders (FWO), Belgium, through the Odysseus programme (Project G0G1316N) and the Akademia grant of Equinor.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematics and Natural SciencesUniversity of BergenBergenNorway
  2. 2.Uni Research CIPRBergenNorway
  3. 3.Faculty of SciencesHasselt UniversityDiepenbeekBelgium

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